Energy eigenfunctions radial

Table 17.2 Radial Factors for Hydrogen-Like Energy Eigenfunctions...

The bound-state energies and eigenfunctions can be obtained by solving the Schrodinger equation with boundary conditions that the radial wave function vanishes at both ends... 

This arbitrariness was noticed by Cohen and Heine and by Schlosser28 who suggested that one way of resolving the difficulty would be to place extra conditions on the pseudopotential operator. In particular they proposed that the eigenfunctions xm should be required to have the minimum possible radial kinetic energy ... 

Figure 7.1 Radial eigenfunctions Pn((r) = rR fr) for the electron in the hydrogen atom (in atomic units) where n is the principal quantum number,

The waves i ° and 0 can be calculated on the dividing surface without difficulty for any collision energy E, because the inter-fragment potential for i > s is purely radial. The internal eigenfunctions of the BC fragment, (pa i i y) i constitute an orthonormal basis on dS [60,158]. In this representation, the matrices and of the incoming... 

The ionization limit of the Schrodinger equation and its eigenfunctions for the free hydrogen atom, at a vanishing energy value, corresponds to Bessel functions in the radial coordinate as known in the literature and illustrated in 2.1. The counterparts for paraboloidal [21], hyperboloidal [9], and polar angle [22] coordinates have also been shown to involve Bessel functions. These limits and their counterparts for the other coordinates are reviewed successively in this section. 

We turn now to the exact determination of these energy shifts. A direct approach through solving the eigenvalue problem numerically, with appropriate nuclear potential V[,uc (r), yields radial function(s) (eigenfunctions) and total energies (eigenvalues). The reliable determination of the... 

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